Find the average value of $0$, $2z$, $4z$, $8z$, and $16z$.
To find the mean of these $5$ values, we must add them and then divide by $5$.  Thus we get that the answer is $\frac{0+2z+4z+8z+16z}{5}=\frac{30z}{5} = \boxed{6z}$.